Approximation and regularization of problems of the optimal control of the coefficients of parabolic equations
The author considers an optimal control with cost functional \[ J(u,g)= \alpha_ 1 \int_ Q | u(x,t)- u_ 0(x,t)|^ 2 dx dt+ \alpha_ 2 \int_ \Omega | u(x,T)- \psi(x)|^ 2 dx \] and state equation with control in the coefficients \[ \begin{cases} u_ t- D_{x_ 1}(g_ 1(x,t)D_{x_ 1} u)- D_{x_ 2} (g_ 2(x,t)D_{x_ 2}u)+ g_ 3(x,t)u= f(x,t) &\text{in }Q\\ u(x,t)= 0 &\text{on }\partial\Omega\times (0,T)\\ u(x,0)= \phi(x) &\text{on }\Omega.\end{cases} \] Here, \(\Omega\) is a rectangle in \(\mathbb{R}^ 2\), \(Q= \Omega\times ]0,T]\), \(f\), \(\phi\), \(u_ 0\), \(\psi\) are given, and \(g= (g_ 1,g_ 2,g_ 3)\) is the control variable which is assumed to satisfy suitable boundedness conditions on the derivatives. An existence result for optimal pairs is then easily proved. The main goal of the paper is to study finite-dimensional approximation schemes for the optimal control problem above, as the difference method or the projection-difference method. A priori bounds are obtained for the error with respect to the state, and the rate of convergence of approximations is estimated.
scientific article; zbMATH DE number 584216
scientific article